When Mrs. Obama was roughly 12 minutes into her 20-minute remarks at a home in Northwest Washington, a woman at the front of a crowd of about 200 people began shouting for President Obama to issue an executive order barring federal contractors from discriminating against employees based on sexual orientation or gender identity. But where Mr. Obama, more accustomed to such interruptions, typically waits in place for the protester to stop and perhaps acknowledges the complaint, his wife chose direct confrontation.